Estimating a Profit Function in the Presence of...

Journal of Agricultural and Resource Economics 27(1):94-113Copyright 2002 Western Agricultural Economics Association

Estimating a Profit Functionin the Presence of Inefficiency:

An Application to Russian Agriculture

Carlos Arnade and Michael A. Trueblood

The relationships among cost functions, distance functions, and technical inefficiencyare utilized to show how technical inefficiency scores can be incorporated into thespecification of a profit function and a related system of output supply and inputdemands. A method also is introduced for incorporating allocative efficiency scoresinto the same system. The theoretical and empirical approach requires fewer assump-tions than those made in many studies. An illustrative example is provided forRussian agriculture for 1994-95, a period when significant technical and allocativeinefficiency was known to exist. The results demonstrate inefficiency limits the supplyresponse to prices, thus leading to lower estimates of output response compared toa traditional supply model in which efficiency is assumed.

Key words: distance function, profit function, Russia, supply response, technicalefficiency

Introduction

In the past two decades, much of the empirical literature concerning agricultural pro-duction has fallen into two major categories. One strand of the literature estimates theprice response of agricultural supply and input demands assuming efficiency (e.g., Ball;Shumway, Saez, and Gottret). Another strand calculates various producer inefficiencies,particularly technical inefficiency (e.g., Battese; Seiford), and ignores price responses.However, few studies in the literature have combined these two issues. The objective ofthis study is to introduce a method for modeling and testing the impact of efficiencymeasures on output supply and input demand.

One method for dealing with both production inefficiencies and the supply responseto prices requires joint estimation of profit function parameters and efficiency scores ina single step (Kumbhakar 1996, 2001). Specific estimates of inefficiency scores for eachobservation are derived from an estimated parameter in the profit function and adistribution function of model errors. While this technique has many advantages, suchas being able to apply statistical testing procedures to estimates of inefficiency, it relieson assumptions about model error structure and can be computationally intensive.Using this technique, it is not always possible to overcome the difficult task of sortingout allocative inefficiency from technical inefficiency.

Carlos Arnade and Michael A. Trueblood are economists with the U.S. Department of Agriculture, Economic ResearchService, Washington, DC. The views expressed here are the authors' and do not necessarily represent those of the EconomicResearch Service or the U.S. Department ofAgriculture. The authors thank David Sedik and Stefan Osborne for their helpfulcomments.

Review coordinated by Gary D. Thompson.

Inefficiency and Supply Response in Russia 95

In this article, a method is developed for representing inefficiency scores as variableswhen estimating a system of output supply and input demand equations.1 Using a novelapproach, this study demonstrates theoretically one way technical and allocative ineffi-ciency can be incorporated into a system of output supply and input demand equations.Our theoretical and empirical approach requires fewer assumptions than those madein many studies, but must be implemented in two steps.

In the first step, inefficiency scores are obtained using programming techniques. Thesecond step employs econometric methods to model the impact these scores have onoutput supply and input demand. To demonstrate the method, technical and allocativeinefficiency scores for 73 Russian oblasts (equivalent to states or provinces) are calcu-lated using standard nonparametric methods. Then, the inefficiency scores are specifiedas an explanatory variable in a profit function for Russian agriculture. A system ofoutput supply and input demand equations is derived from this profit function. Usingtwo years (1994 and 1995) of cross-sectional data, the parameters of the system of out-put supply and input demands are estimated econometrically, taking into account theeffect inefficiency scores may have on parameter estimates.

To test the alternative model specifications in the Russian example, standardeconometric tests are used to determine the contribution of inefficiency scores to theperformance of the estimated model. Statistical tests of the contribution of technical andallocative inefficiency measures to model performance are mixed. The model showsRussian producers were sensitive to some prices, particularly input prices in the 1994-1995 period, and widespread technical and allocative inefficiencies restricted theresponse of output supply to price changes. While these findings provide some insightinto Russian agriculture, the empirical section primarily serves the goal of demonstratingone method for estimating output supply and input demand in the presence of technicaland allocative inefficiency.

Deriving a Profit Functionin the Presence of Inefficiency

Kumbhakar (1996) provides one of the most comprehensive examples for jointly esti-mating inefficiency measures and dual behavioral functions. To derive the profit function,inefficiency was specified as interacting with inputs in a production function. Kumbhakar(1991) used a parallel approach to specify and estimate cost functions in the presenceof inefficiency. The production function with an inefficiency term was specified as partof a standard profit-maximization problem to derive the corresponding profit function.The profit function was specified as a function of shadow prices, and an inefficiencyparameter was estimated jointly with the profit function parameters. Observation-specific inefficiency scores were then obtained from this inefficiency parameter and thedistribution functions of the econometric error.

Kumbhakar's technique has the advantage of estimating inefficiency scores andeconometric parameters in one step. Furthermore, the approach allows standardstatistical testing procedures to establish a level of confidence in the inefficiency scores.

1 Jonasson and Apland have estimated an agricultural sector supply model that first estimates farm-level inefficiency asan input into a sector supply model. In their study, inefficiency and supply response were both calculated using linear pro-gramming methods.

Arnade and Trueblood

Journal ofAgricultural and Resource Economics

However, this methodology relies on computationally intensive estimation techniques,

cannot always distinguish different types of inefficiency, and imposes restrictions on the

distribution of model errors. In contrast, the method introduced here requires two steps

and does not subject inefficiency scores to statistical tests. Further, our approach relies

on less restrictive assumptions and sorts out the effects of technical and allocative ineffi-

ciency.

Technical Inefficiency

Economists may be less interested in using statistical procedures to test inefficiencyscore estimates than in using statistical procedures to test the impact inefficiency hason output supply or input demand. There are several possible ways to model and test

the effect of technical inefficiency on producer behavior. The approach developed below

exploits the established dual relationship between cost and distance functions, and theestablished relationship between distance functions and technical inefficiency. Thesetwo previously established relationships are joined to show there already exists a rela-tionship connecting the primal concept of technical inefficiency to the specification of a

profit function, and by extension, the supply function.To observe this, consider a production technology that is homogeneous of degree k in

inputs and outputs, and in which outputs are separable from inputs. One of the numer-ous options used for characterizing the technology is the input distance function (Fareand Primont, p. 152). By definition, the input measure of technical inefficiency, whichrepresents the amount by which inputs can be reduced to produce the same level of

output, is equal to the reciprocal of the input distance function (e.g., Fare, Grosskopf,

and Lovell). The input distance function is homogeneous of degree - 1/k in outputs if the

technology is homogeneous of degree k (Fare and Primont), i.e.,

(1) -D(y, x) = D I(yy, = ),

where DIO() is the input distance function, y is an n-vector of outputs, x is an m-vector

of inputs, and y is a parameter.There is a duality between the cost function and any function representing the frontier

of the technology. The duality between the input distance function and cost function

when there is no inefficiency present is specified as:2

(2) C(y, w) = min w'x, s.t.: DI(y, w) = 1,x

where w is a vector of input prices, and the other terms are as previously defined.Suppose there exists a firm which is technically inefficient, and the measure of ineffi-

ciency is 0. As established by Fare et al., the distance function is equal to the reciprocalof programming-based measures of the firm's technical inefficiency. This relationship

indicates that if technical inefficiency is present, the behavior of the firm is influenced

by this inefficiency. To represent this, the cost-minimization problem may be written as

follows:

2 See Shepard or Deaton for an analogous discussion of the expenditure function.

96 July 2002


(3) min w'x, s.t.: DI(y, x) =x 0

= min w'x, s.t.: 0DI(y, x) = 1

= min w'x, s.t.: D(06-ky, x) = 1

= C(0-ky, w) = 0-1C(y, w).

The transition between the second and third lines of (3) comes from the degree of homo-geneity of the distance function. The last line results from the properties of cost functionsunder output homogeneous technology (Chambers; Fare and Primont).

The corresponding profit-maximization problem can be denoted by:

(4) max p'y - 0-OC(y, w),y

where p is a conformable vector of output prices.The first-order condition for each yi is:

aC(5) Pi , i = 1,...,N.

HyiThe resulting profit function can be written as:

(6) l*(Op, w) = max p'y* - 0-1C(y*, w),

where y* denotes the optimal output levels. Taking the derivative of (6) with respect toprices, it can be shown Hotelling's lemma still holds so that:

(7) i = i*(0 p, w), i = 1, ... ,N,apian*

- = x*(Op, w)/O, j = 1,...,M,aWj

where yi* is output of good i, and xJ* is the total amount of inputj used. Furthermore, byapplying envelope properties and the quotient rule, we can show:

(8) Ia n _ C(y,w)ae 02

Using cost data as the endogenous variable, equation (8) can be estimated jointly witha system of supply and input demand equations in (7). However, in models with severalinputs and outputs, multicollinearity likely would be a problem in estimating equation(8).

The attraction of the above specification is twofold. First, economic theory is used todetermine how technical inefficiency scores may be incorporated into a profit function.Second, the procedure shows that if inefficiency scores are available prior to econometricestimation, they can be specified as an exogenous variable in an output supply and inputdemand system, and standard econometric methods can be used to estimate the param-eters of this system. 3

3Kumbhakar (1996) discusses the benefits and problems parametric methods have had dealing with both technical and

allocative efficiency in the context of dual functions.



Allocative and Technical Inefficiency

Unlike technical inefficiency, allocative inefficiency is a behavioral concept arising whenagents do not exactly meet first-order conditions. Toda was one of the first economiststo address the issue of allocative inefficiencies on dual functions when he estimated acost function for Soviet manufacturing industries.

In this section, allocative inefficiency is represented as an additive distortion to first-order conditions. This procedure implies that even when the distortion is the same foreach first-order condition, output (input) price ratios fail to equal the marginal rate oftransformation (substitution). To see this, consider the profit-maximization problem inequation (4). When there is allocative inefficiency, the first-order condition for each yjis:

(9) Pi= 0- 1 aC +ai, i,,N,ayi

where ai represents inefficiency in allocating product i. Rewriting (9) for each yi gives:

(10) O(pi - ai) =-ayi

By (10), even if each ai were the same, the allocative distortion would change the relativeprice ratios. Therefore, it is assumed behavioral mistakes do not neutralize each other.The resulting profit function is represented by:

(11) nI*(0(p - a), w) = max p'y* - 0-1C(y*, w).y

Thus, in this model, technical inefficiency interacts with output prices multiplicatively,while allocative inefficiency interacts additively. Taking the derivative of profits withrespect to the price of output 1, for example, results in the following:

(12) all*(0(p - a), w) + _ aC(y*, w) ayi ap(12) Y + Pi - y, iap i=1 ayi ap; ap1

where p; = 0(p1 - al).When allocative inefficiency exists, output prices do not equal marginal cost divided

by technical inefficiency. This prevents the indirect terms in equation (12) from droppingout. However, using equation (9) and substituting a, into the difference term in paren-theses, we obtain:

aH*(O(p - a), w) , ay(13) -Y 1 + EaiOadP i=l ap 1

since all ap*/api = 0.The above conditions hold for all output price derivatives. Similarly, the derivatives

with respect to changes in input prices are given by:

(14) alJ*(O(p - a), w) (xi + 0 ay(14) ( ) Eaaw. j=1 aWj

The output supply and input demands in (13) and (14) can be easily estimatedincluding the ai O(ay/lpi*) term with the derivative of the profit function. The paramet

98 July 2002


making up the subcomponent term, (ay/9pi*), can be derived analytically from the supplyand demand functions.

An Empirical Application:Russia During Transition

Data

To apply this model, data are used from 73 Russian oblasts (equivalent to states or prov-

inces) for 1994-95. While Russia's transition to a market economy has been ongoing, the

years 1994 and 1995 represented a unique period when the agricultural sector faced

market prices, but retained elements of a planned economy. Studies of this period have

demonstrated that technical inefficiencies were prevalent in Russian agriculture (Brock;

Sotnikov; Sedik, Trueblood, and Arnade).Oblast-level prices and quantities of crops or crop categories, as well as the prices and

quantities of inputs used in crop production of corporate (i.e., state and collective) farms,are all taken or derived from official sources (Russian Ministry of Agriculture; Goskom-

stat of Russia). Corporate farms produce approximately half the value of Russian

agriculture, and account for approximately 90% of grain and sugar beets production and

slightly less than half the livestock production. A significant amount of Russian agricul-

tural output is produced on small private plots. Production from private plots is mostly

not included in these data.The output data consist of quantities and prices for five production categories: grains,

sugar beets, potatoes, vegetables, and livestock. The input quantity data consist of land,labor, fertilizers, oil, fuels, electricity, and tractors. Input price data were available for

labor, fertilizers, oil, fuels, and electricity. Labor, oil, fuels, electricity, and fertilizer were

modeled as variable inputs, while tractors and land were modeled as quasi-fixed inputs.

The output quantities were all measured in tons per year. The input quantities were

measured as follows. Land was measured as land sown to the four crop categories on

corporate farms. Labor was measured in 1,000 man-days worked in production. Ferti-

lizers were measured in 1,000 tons and were derived from corporate farm purchases

using farm expenditures and unit prices. Oil and fuel products, measured in tons, alsowere derived from purchases by corporate farms using a similar procedure. Electricity,

primarily used for hothouse production in northern regions, was measured as 1,000-

kilowatt hours from expenditure and unit prices. Tractors were measured in 1,000 HP

units.The output prices were farm gate prices and measured in 1,000 rubles per ton. The

grains and livestock prices were weighted average prices in each oblast, where the

weights were derived from the value of output of each subcomponent in the category.Input prices were measured in the following way. For labor, average daily wages were

calculating by dividing monthly salaries by the number of man-days worked in the month(Russian Ministry of Agriculture). Fertilizer, oil, and fuel prices were available from

price publications (Goskomstat of Russia). The published input prices are weighted aver-

age prices in each oblast, where the weights are proportional to total input purchasesor output sold on each farm in the oblast. For both output and input prices, the centralstatistical agency, Goskomstat, estimates prices from a sample of 10% of all agriculturalenterprises in Russia.


Journal of Agricultural and Resource Economics

Output Quotas and Specification of a Profit Function

An important institutional constraint to be addressed was the possible existence ofoutput quotas during the sample period. Under traditional Soviet agriculture, outputquotas often were set for agricultural production units (Brock). Arnade and Munisamyconcluded Russian agricultural producers in the early post-Soviet period still might havebeen prone to output targeting. They applied data envelopment analysis (DEA) to a profit-maximizing problem and tested for broad targeting of outputs in Russian agricultureusing a procedure similar to that used to test for expenditure constraints (Fare, Gross-kopf, and Lee). In their analysis, however, Arnade and Munisamy did not attempt todetermine which crops might have been targeted for quotas.

To specify a profit function for Russian agriculture, the approach employed by Arnadeand Munisamy was modified to test for the influence of crop-specific output constraints.The tests for output targeting were applied to each of the 73 Russian oblasts for 1994,and then again to the same oblasts in 1995. DEA profit-maximization problems weresolved with and without output constraints, and then solution profits and actual profitswere compared.

From this comparison, an index was created: a value less than one indicated outputtargets limited profits, while an index measure close to one implied observed levels ofoutput were consistent with the objective of profit maximization (see Arnade and Muni-samy). For sugar beets, the average index measure across all oblasts was 0.96, with 48oblast regions registering a one. Similarly, for grains, the average across all oblasts was0.95, with 46 oblast regions registering a one. Averages for potatoes and vegetables,however, were 0.56 and 0.67, respectively, with no region registering a one. Based onthis information, we determined vegetables and potatoes should be treated as fixed out-puts, while grains and sugar beets should be specified as variable outputs in the profitfunction for Russian agriculture.4

Having established that potatoes and vegetables were fixed outputs, a normalizedquadratic functional form was specified. The normalized quadratic form satisfies theproperties of the profit function (Shumway, Saez, and Gottret), is flexible, and yet avoidsthe computational burdens imposed by other functional forms. The following normalizedquadratic profit function was specified:

(15) H(Op, w) = E i0(i - ai) + E Iwj1 + 0.5 E E P(Pi - a)(Pk - ak)i=l j=1 i=l k=l

4 4 3 4

+ 0.5 E PkWWk + E Yid0(Pi - ai)LND + E yidwjLNDj=l 1=1 i=1 j=l

2 3 2 4

+ E ziQzO(Pi - ai) + E yz QzWjz=1 l z=l j=l

3 4 3 4

+ 0.5 E E Piko(Pi - ai)Wk + yiiTRC + E yJTRCi=l k=l i=l j=l

The output prices represent aggregate grains, sugar beets, and aggregate livestock. Theprices of variable inputs, wj, represent fertilizer, fuel, electricity, and oil. All prices were

4 Vegetables and potatoes may be targeted for household food security concerns.

100 July 2002


normalized by the price of fertilizers. The fixed outputs, Q, as noted earlier, are vege-tables and potatoes. TRC represents tractors and LND denotes land, both quasi-fixedinputs. In the Russian model, each data point represents one oblast observation in oneof the two periods (1994 and 1995).

Using the relationship in equation (13), the ith supply equation is specified as:

3 4

(16) Yi = Pie + E 2(Pk - ak) + E PijWk + yit O TRC + YidOLNDk=l j=l

2 3

+ YizOQz E PkiOakz=l k=l

Factoring the ak frompk in the second term on the right-hand side of (16) and using sym-metry (Pik = Pki), the equation to be estimated becomes:

3 3 4

(17) Yibt = Pi bt - Pikak(Obt + Ot) + E Pik bt + E PiejWk=l k=l j=l

2

+ yitOTRCbt + YidOLNDbt + ̂ izOQzbt + 6 ibt,z=l

where the outputs are represented by {i = 1, 2,3}, the oblasts are denoted by {b = 1,..., 73},and t = 1994, 1995.

Similarly, using equation (14), the input demands are calculated as:

4 3 4

(18) -Xjbt = Pabt E Pjkak(Obt + Obt) + E Pjio Pibt +Pibt + E PkWkbtk=l j=l k=l

2

+ YZjOTRCbt + yjdOLNDbt + E Yjz Qzbt + jbt, j = 1,..., 4.z=l

The e terms appended to equations (17) and (18) represent random errors, which areassumed to be independent and identically distributed normal random variables withzero means and constant covariances:

(19) E{ ibt} = E{ejbt = 0,

E{ ibt, Eibt} =ii

(20) E{:jbt, ejbt] = °jj for b = d and t = s; 0 otherwise.

Ee{ ibt E jbt = O i

These assumptions are reasonable and commonly used. They allow for contemporaneouscorrelation of the errors from different supply and input demand equations within eachoblast unit.

Equations (17) and (18) are supply equations which account for both technical andallocative inefficiency. When producers are technically efficient, 0 = 1, and when theyare allocatively efficient, a = 1. If all producers are both technically and allocatively effi-cient, then equations (17) and (18) reduce to standard output supply and input demandequations.



Inefficiency Calculations

The techniques widely applied to measuring firm or regional inefficiency fall into twobroad categories: (a) the parametric stochastic frontiers approach, which was separatelyintroduced in 1977 by Aigner, Lovell, and Schmidt, and by Meeusen and van den Broeck,and (b) the nonparametric programming approach, introduced in 1957 by Farrell, andsubsequently revived as data envelopment analysis (DEA) by Charnes, Cooper, andRhodes in 1978.

Both classes of models have seen considerable refinement and widened applicationsover the past 20 years. While each technique has its own advantages and limitations(e.g., see Hjalmarsson, Kumbhakar, and Heshmati), technical inefficiency measures cal-culated with nonstochastic programming methods can be used as explanatory variablesin subsequent econometric analysis without sequential econometric estimation. Thus weuse a DEA problem in this analysis to calculate technical inefficiency measures for 73oblasts across Russia. Statistical tests are not used to determine the level of confidencein each inefficiency score. However, in the next section, standard testing procedures areused to determine the impact these scores have on output supply and input demand.

In estimating technical inefficiency, a procedure similar to that reported by Fare andWhittaker is followed in which a model is developed for a farm producing livestock andcrops, with the crop output used as on-farm livestock feed (see Fare and Whittaker fordetails). Technical inefficiency in this study is estimated at the oblast level, and it isassumed corporate farms in each oblast have similar technology. This model also allowscrop output to be represented in two components: (a) output used as feed within theoblast, whether through sales or barter arrangements, and (b) output sold outside theoblast. The feed constraint of the Fare and Whittaker model was modified here to accountfor feed imported by the oblast. Using similar assumptions, a DEA program also was usedto calculate allocative inefficiency scores for each oblast (see Coelli, Rao, and Battese).

In calculating technical inefficiency scores, no assumptions about scale economieswere made. Instead, inefficiency scores were used to determine the scale economy ineach oblast. By comparing the inefficiency scores estimated with different scale assump-tions, Byrnes et al. demonstrated it is possible to determine whether there is decreasing,constant, or increasing returns to scale. To make this determination, a program was runthree times for each observation-one time each assuming constant (CRS), decreasing(DRS), and variable returns to scale (VRS). In Byrnes et al.'s method, if a particular scoreis equal when calculated across all scale assumptions, then the observation is consideredto be CRS. If DRS and CRS scores are not equal, then the observation is considered to beDRS. If the inefficiency score is equal when calculated under DRS and CRS assumptions,but not equal to the score calculated under the VRS assumption, then the technology isconsidered to be increasing returns to scale (IRS).

Table 1 reports the calculated technical inefficiency scores under the various scaleassumptions, and table 2 presents calculated allocative inefficiency scores. Comparingtechnical inefficiency scores, agricultural technology at the oblast level in Russia wasfound to be more commonly characterized by decreasing returns to scale than constantor increasing returns. For example, in 1995, 51 oblasts displayed decreasing returns toscale, 15 showed constant returns, and seven displayed increasing returns.5 In general,

5 Sedik, Trueblood, and Arnade found increasing returns for the crops using a sample for 70 oblasts. The present studydiffers in that it estimates returns to scale on an oblast-by-oblast basis, uses an alternative method, and examines returnsto scale for both crops and livestock.

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Table 1. Technical Inefficiency Estimates for 73 Russian Oblasts, by Region:Joint Crop and Livestock Model (1994 and 1995)

Regions/Oblasts

1995

VRSa CRS DRS Scaleb

1994

VRS CRS DRS' Scaleb

North:ArkhangelskKareliaKomiMurmanskVologda

Northwest:LeningradNovgorodPskov

Central:BryanskIvanovskKaliningradKalugaKostromaMoscowOrlovRyazanSmolenskTulaTverVladimirYaroslavl

Volga:ChuvashiaKirovMari-ElMordoviaNizhniy Novgorod

Central Black Soil:BelgorodKurskLipetskTambovVoronezh

Volga Valley:AstrakhanKalmykiaPenzaSamaraSaratovTatarstanUlyanovskVolgograd

1.000.640.831.000.70

0.680.630.831.000.45

1.000.640.831.000.70

DRSDRSCRS

CRSDRS

0.661.000.831.000.70

0.490.950.751.000.39

0.661.000.831.000.70

DRSDRSDRS

CRSDRS

0.92 0.53 0.92 DRS 1.00 0.59 1.00 DRS0.45 0.45 0.45 CRS 0.39 0.38 0.39 DRS0.38 0.37 0.38 DRS 0.40 0.39 0.39 IRS

0.400.400.460.350.411.000.380.580.330.600.320.560.41

0.580.390.460.380.58

0.670.540.390.730.58

1.000.380.400.851.000.970.530.50

0.340.300.460.320.400.590.260.410.310.330.310.470.41

0.400.280.430.370.36

0.360.330.220.670.32

1.000.370.380.690.910.380.430.48

0.400.400.460.350.411.000.380.580.330.600.320.560.41

0.580.390.460.380.58

0.670.540.390.730.58

1.000.370.400.851.000.970.530.50

DRSDRSCRSDRSDRSDRSDRSDRSDRSDRSDRSDRSCRS

DRSDRSDRSDRSDRS

DRSDRSDRSDRSDRS

CRSIRSDRSDRSDRSDRSDRSDRS

0.640.530.640.650.431.000.580.700.500.720.540.610.58

0.600.520.560.610.65

0.730.730.670.930.65

0.650.840.730.841.001.000.560.92

0.470.450.580.540.370.820.480.480.350.570.390.450.49

0.530.310.480.550.45

0.510.560.580.630.41

0.640.840.520.601.000.850.480.70

0.640.530.640.650.431.000.580.700.500.720.540.610.58

0.600.520.560.610.65

0.730.730.670.930.65

DRSDRSDRSDRSDRSDRSDRSDRSDRSDRSDRSDRSDRS

DRSDRSDRSDRSDRS

DRSDRSDRSDRSDRS

0.64 IRS0.84 CRS0.73 DRS0.84 DRS1.00 CRS1.00 DRS0.56 DRS0.92 DRS

(continued...)


Journal of Agricultural and Resource Economics

Table 1. Continued

1995

Regions/Oblasts VRS a CRS a DRS a Scale b

North Causcus:A - _ A _I A , ,_

1994

VRSa CRS DRS' Scale b

Adygea 0.55 0.44 0.44 IRS 0.73 0.67 0.73 DRSDagestan 0.81 0.78 0.81 DRS 0.56 0.53 0.56 DRSKabardino-Bal. 0.58 0.53 0.53 IRS 0.69 0.65 0.69 DRSKarachay-Cher 0.73 0.53 0.53 IRS 0.73 0.72 0.72 IRSKrasnodar 1.00 0.56 1.00 DRS 1.00 0.78 1.00 DRSNorth Osetia 0.71 0.49 0.49 IRS 0.92 0.74 0.74 IRSRostov 1.00 0.94 1.00 DRS 1.00 0.67 1.00 DRSStavropol 1.00 0.68 1.00 DRS 0.93 0.70 0.93 DRS

Urals:Bashkortostan 0.70 0.27 0.70 DRS 0.55 0.45 0.55 DRSChelyabinsk 0.56 0.50 0.56 DRS 0.82 0.59 0.82 DRSKurgan 0.48 0.40 0.48 DRS 0.77 0.57 0.77 DRSOrenburg 0.35 0.34 0.35 DRS 0.89 0.71 0.89 DRSPerm 0.85 0.51 0.85 DRS 0.82 0.47 0.82 DRSSverdlovsk 1.00 1.00 1.00 CRS 1.00 0.48 1.00 DRSUdmurtia 0.55 0.36 0.55 DRS 0.61 0.43 0.61 DRS

West Siberia:Altay Krai 0.85 0.54 0.85 DRS 0.81 0.66 0.81 DRSAltay Republic 1.00 1.00 1.00 CRS 1.00 1.00 1.00 CRSKemerova 0.73 0.41 0.73 DRS 1.00 0.52 1.00 DRSNovosibirsk 1.00 1.00 1.00 CRS 0.78 0.56 0.78 DRSOmsk 0.81 0.76 0.81 DRS 1.00 0.91 1.00 DRSTomsk 0.80 0.49 0.80 DRS 0.56 0.45 0.56 DRSTyumen 1.00 0.61 1.00 DRS 0.74 0.44 0.74 DRS

East Siberia:Buryatia 0.59 0.55 0.59 DRS 0.62 0.62 0.62 CRSChitinsk 0.57 0.57 0.57 CRS 0.54 0.53 0.53 IRSIrkutsk 0.60 0.31 0.60 DRS 0.91 0.60 0.91 DRSKhakassia 0.37 0.35 0.35 IRS 0.39 0.38 0.38 IRSKrasnoyarsk 0.84 0.35 0.84 DRS 0.91 0.50 0.91 DRS

Far East:Amur 0.47 0.46 0.47 DRS 0.78 0.59 0.78 DRSKamchatka 1.00 1.00 1.00 CRS 1.00 1.00 1.00 CRSKhabarovsk 0.99 0.98 0.98 IRS 1.00 0.84 1.00 DRSMagadan 1.00 1.00 1.00 CRS 1.00 1.00 1.00 CRSPrimorye 1.00 1.00 1.00 CRS 1.00 0.84 1.00 DRSSakhalin 1.00 1.00 1.00 CRS 1.00 1.00 1.00 CRSYakutia 1.00 1.00 1.00 CRS 1.00 1.00 1.00 CRS

aVRS imposes variable returns to scale, CRS imposes constant returns to scale, and DRS imposes decreasingreturns to scale.b Scale: By imposing VRS, CRS, and DRS, it is possible to infer which oblast technology is CRS, IRS, or DRS.

both technical and allocative inefficiency displayed much variation, particularly in theCentral and Volga valley regions.

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Table 2. Allocative Inefficiency Scores for 73 Russian Oblasts, by Region(1994 and 1995)

Regions/Oblasts 1995 1994 Regions/Oblasts

North:ArkhangelskKareliaKomiMurmanskVologda

Northwest:LeningradNovgorodPskov

Central:BryanskIvanovskKaliningradKalugaKostromaMoscowOrlovRyazan

SmolenskTulaTverVladimirYaroslavl

Volga:ChuvashiaKirovMari-ElMordoviaNizhniy Novgorod

Central Black Soil:BelgorodKurskLipetskTambovVoronezh

Volga Valley:AstrakhanKalmykiaPenzaSamaraSaratovTatarstanUlyanovskVolgograd

0.670.430.481.001.00

0.770.180.19

0.320.13

0.250.180.261.00

0.26

0.22

0.170.410.22

0.280.27

0.51

0.23

0.34

0.280.33

0.56

0.430.14

0.640.46

1.00

0.370.280.481.00

0.71

0.38

0.48

0.33

1.00

0.82

1.00

1.00

1.00

0.31

0.39

0.19

0.26

0.64

0.40

0.43

1.00

0.47

0.33

0.49

0.73

0.54

0.48

0.58

0.60

0.48

0.56

0.61

0.65

0.73

0.73

0.31

0.93

0.65

0.65

0.84

0.58

0.84

1.00

0.54

0.46

0.91

North Causcus:AdygeaDagestanKabardino-Bal.Karachay-CherKrasnodarNorth OsetiaRostovStavropol

Urals:BashkortostanChelyabinskKurganOrenburgPermSverdlovskUdmurtia

West Siberia:Altay KraiAltay RepublicKemerovaNovosibirskOmskTomskTyumen

East Siberia:BuryatiaChitinskIrkutskKhakassiaKrasnoyarsk

Far East:AmurKamchatkaKhabarovskMagadanPrimoryeSakhalinYakutia

1995 1994

1.00

0.45

0.15

0.35

0.54

0.48

0.40

0.48

0.47

1.00

0.38

0.21

0.37

1.00

0.19

0.49

1.00

0.49

1.00

0.80

0.34

0.75

0.36

0.32

0.36

0.27

0.50

0.16

0.83

0.76

0.87

0.35

0.25

0.84

0.73

0.56

0.26

0.73

1.00

0.92

0.58

0.55

0.54

1.00

0.74

0.60

0.82

1.00

0.42

0.69

1.00

1.00

0.77

1.00

0.56

0.74

0.62

0.53

0.32

0.90

0.91

0.45

1.00

1.00

0.87

0.45

0.25

0.84



Estimation of Output Supply and Input Demandwith Technical and Allocative Inefficiency

While the existence of technical and allocative inefficiency is not unique to Russia, itscritical role in the agricultural economy of Russia may be unique where, despite a moveto market pricing, institutional constraints and poor management practices persist(Brock; Sotnikov; Sedik, Trueblood, and Arnade). Thus, Russian agriculture is appropri-ate for demonstrating this investigation's modeling technique.

Having calculated technical and allocative inefficiency, three different models of out-put supply and input demand were estimated. Model 1 is a standard model in which fullefficiency is assumed, model 2 relaxes the efficiency assumption by allowing technicalinefficiency, and model 3 relaxes the assumption further to include both technical andallocative inefficiency.6 In models 2 and 3, technical inefficiency scores representing theappropriate degree of scale (CRS, DRS, or IRS) in each oblast were used. Thus, an exog-enous variable is used to account for differences in scale economies in each observation.

In each model, cross-sectional data representing 73 oblasts for 1994-95 were used toestimate a seven-equation system (three output supply and four input demand equa-tions) with iterative seemingly unrelated regression (ITSUR) with correction for censoreddependent variables, which is equivalent to maximum-likelihood estimation. A dummyintercept variable was used to represent a change in time periods.7 Relative outputprices were represented by lagged relative output prices, which implies producers follownaive relative price expectations. Lagging relative output prices reduced the data set totwo years (1994, 1995) and 73 cross-sections, but provided the model with an acceptablerepresentation of expected output price. Standard symmetry conditions were imposedin the second-stage ITSUR estimation.

A problem encountered in the econometric estimation was that several Russianoblasts did not grow some of the crops. For example, a few oblasts did not grow grainsand about one-third did not thgrow sugar beets. This problem was addressed by employinga technique developed by Shonkwiler and Yen for modeling censored dependent variablesin a system of equations. First, probit models were estimated for the two dependent var-iables (sugar beets and grains) containing the zero observations. Then, using the approp-riate information from the probit models, and restricting the parameters on the probitmodel to equal their first-stage estimates, the probit equations were included in thesystem of output supply and input demand equations, which were jointly estimated withthe ITSUR procedure.

The estimated parameters of model 2 are presented in table 3 (for brevity, parametervalues of output supply and input demand for models 1 and 3 are not shown). 8 The own-price coefficients of model 2, which include technical inefficiency scores, are statisticallysignificant at the 0.05 confidence level in three out of four input demand equations, andin one out of three output supply equations. In both the grains and sugar beet equations,the cross-price coefficients between these two crops are statistically significant. The

6 Note, because the profit function maximizes a linear profit equation, the various uses of grain for on-farm feed need notbe broken out in a dual problem.

7 Distinct output supply and input demand functions for each period could have been estimated, but to keep the presen-tation of results simple, all available data were used to estimate a single system of equations.

8 The tables for models 1 and 3 are available upon request from the authors. In model 1, all own-price elasticities displayedthe expected signs. In model 3-the model with both allocative and technical inefficiency-the parameters were not muchdifferent from those of model 2.

106 July 2002


Table 3. Output Supply and Input Demand for 73 Russian Oblasts, Estimatedby ITSUR Corrected for Censoring (1994-1995)

OUTPUTS INPUTS

Variable Grains Sugar Livestock Labor Oil Electricity Fuel

02*PGRN 138,820.0 -127,141.1 -21,109.0 -234.6 -443.6 50,028.1 208,600.0(0.82) (-1.72) (-0.60) (-0.15) (-0.92) (0.35) (2.93)

02*PSUG -127,141.1 161,910.0 25,491.0 800.5 26,838.0 25,052.0 26,875.0(-1.72) (1.92) (1.29) (0.97) (1.20) (0.34) (0.71)

02 *PLVE -21,109.0 2,591.0 2,674,500.0 -107,270.0 8,672.3 20,117.0 25,130.0(-0.60) (1.29) (0.68) (-0.79) (0.75) (0.27) (0.51)

*WAGES 234.6 -800.5 107,270.0 -16,971.0 354.7 -2,230.2 -3,555.4(0.15) (-0.98) (0.79) (-1.71) (0.61) (-0.76) (-1.86)

*WOIL 443.1 -26,838.0 -8,672.3 354.7 -16,031.0 18,536.0 39,217.0(0.01) (- 1.20) (-0.74) (0.61) (-1.08) (0.41) (1.73)

*WELEC -50,028.3 -25,052.3 -20,117.1 -2,230.2 18,536.0 -1,714,300.0 485,690.0(-0.34) (-0.33) (-0.27) (-0.75) (0.41) (-4.67) (3.39)

o *WFUEL -208,600.1 -26,875.0 -25,130.0 -3,555.5 39,217.1 485,690.5 -1,108,800.0(-2.94) (-0.71) (-0.51) (-1.86) (1.73) (3.39) (-9.67)

O*TRACT 642.7 324.6 98.4 6.67 75.5 97.2 25.7(13.21) (2.73) (11.51) (16.9) (6.06) (2.51) (1.31)

0*LAND -548.1 -693.6 -170.3 -7.48 -61.2 -84.2 -84.0(-4.77) (-2.94) (-8.40) (-7.98) (-2.07) (-0.91) (-1.81)

0*QPOT -0.73 -0.65 0.13 -0.008 0.15 0.72 0.55(-3.68) (-1.03) (3.57) (-4.69) (2.98) (4.48) (1.20)

*QVEG -4.83 -0.31 -0.21 -0.002 -0.28 -1.06 0.36(-6.58) (-0.16) (-1.58) (-2.92) (-1.51) (-1.79) (1.21)

0 -353,430.0 -95,414.0 -98,536.0 -10,686.0 -183,460.0 -306,060.0 -327,300.0(-2.36) (1.65) (-3.06) (-7.99) (-5.00) (-2.05) (-4.68)

YEAR -0.0001 95,414.0 -25,535.0 -31.0 212,460.0 -450,130.0 -38,349.0(-0.50) (1.65) (-2.04) (-0.06) (12.36) (-7.88) (-1.34)

Notes: Numbers in parentheses are t-statistics. The system log-likelihood value is -12,967.3. Single equationmeasures of fit are not generally applicable in systems estimation.a 0 represents the square root of the inefficiency variable, 02; both are used as interaction terms with other vari-ables. PGRN, PSUG, and PLIVE are the respective prices of grain, sugar, and livestock, which are normalized onthe price of fertilizer. Relative output prices were lagged one period to represent naive relative price expectations.WAGES, WOIL, WELEC, and WFUEL are the respective input prices for labor, oil, electricity, and other non-oil fuels.QPOT and QVEG are the fixed output quantity potatoes and vegetables. YEAR is a dummy variable with a valueof 0 for 1994 and 1 for 1995.

coefficient on the one "stand-alone" technical inefficiency term in each equation, 0, is sta-tistically significant in all the equations. Output supply parameters were small and statis-tically insignificant, while the input demand parameters were statistically significant.

The implied own-price elasticities for all three models were calculated at the samplemean for the two-year period and are reported in table 4. Model 2's elasticities, the modelwith technical inefficiency included, are larger in absolute value than those for model1. Further, more of model 2's elasticities are statistically significant at the 0.01 confi-dence level compared to the findings for model 1.9 Model 2's own-price elasticities have

9 Model 1 (standard model) elasticity estimates would be lower if the true model included efficiency scores. By excludingthe efficiency variable, the inefficiency effect may become incorporated into the estimated parameter of price response,resulting in smaller elasticity estimates.



Table 4. Own-Price Elasticities of Models With and Without Technical andAllocative Inefficiency (1994-1995)

MODELS

[#1] [#2] [#3]With No With With Technical

Inefficiency Technical and Allocative(Standard Model) Inefficiency Inefficiency

Description Elasticity Std. Error Elasticity Std. Error Elasticity Std. Error

Outputs:Grains 0.023 0.059 0.055 0.066 0.086 0.066Sugar Beets 0.025 0.061 0.217** 0.112 0.268** 0.122Livestock 0.036 0.036 0.036 0.067 0.026 0.057

Inputs:Wages -0.041 0.077 -0.062** 0.030 -0.053** 0.002Oil -0.029 0.078 -0.091 0.080 -0.090 0.052Electricity -0.356** 0.121 -0.627** 0.130 -0.630** 0.259Fuel -0.488** 0.095 -0.841** 0.173 -0.860** 0.117

Notes: Double asterisks (*) denote significance at the 0.05 confidence level. Cross-elasticities are not shown forthe sake of brevity, but are available upon request from the authors.

the expected signs for grains, sugar beets, and livestock (0.055, 0.217, and 0.036, respec-tively), and indicate there is a slight sensitivity to output prices; the elasticities for grainsand livestock are statistically indistinguishable from zero. All input elasticities have theexpected negative signs, and several show a clear sensitivity to price (electricity, - 0.627;fuels, -0.841). Model 3's output supply elasticities are slightly higher for grains andsugar beets compared with those of model 2 (technical inefficiency only), but the inputdemand elasticities are practically the same as in model 2. Except for electricity, theelasticity estimates from model 3 are more precise.

The positive response to the price of sugar beets and strong negative response to theprices of electricity and fuel provide evidence confirming Russian producers wereresponding to some market signals after three years of market pricing. The signs forboth the output supplies and input demands were as expected theoretically, though thegreatest sensitivity was shown for the demand for electricity and fuels. It makes sensethat changes in electricity and fuel prices elicited a stronger response because these twoinputs were available to substitute for each other. When interpreting these results, itis important to consider: (a) cross-sectional supply models often do not perform as wellas time-series models, and (b) evidence of response to market prices in Russia, even ifit is a relatively small response, is of significant interest to policy makers and transitioneconomy researchers.

A Comparison with a Standard Model

Two different tests were employed to compare model 2 with model 1, and model 3 withmodel 2. To test model 2 (technical inefficiency model) against model 1 (a standardmodel), a nonnested test for systems estimation devised by Davidson and MacKinnonwas used. However, to test model 3 (technical and allocative inefficiency model) againstmodel 2 (technical inefficiency model), a system log likelihood ratio test was used becausemodel 2 was nested within model 3.

108 July 2002


Table 5. System P1 Tests (following a t-distribution)

[A] [B]Ho: Model 1 H0 : Model 2

(standard model) (tech. inefficiency model)Description is the true model is the true model

P1 Value 17.23*** 11.24***

Coefficient 0.54 0.11

Notes: Triple asterisks (*) denote significance at the 0.01 confidence level. The signif-icant P1 statistic in column [A] rejects the model without technical inefficiency; thesignificant P1 statistic in column [B] rejects the model with technical inefficiency. Thecoefficient represents the degree to which the test moves away from the model. Thecoefficient 0.54 means implicitly that it attaches close to equal weights to bothmodels in the compound model, while the coefficient 0.11 attaches a weight of 0.89to the efficiency model and 0.11 to the no-efficiency model.

For the first test comparing model 1 with model 2, Davidson and MacKinnon's methodfor testing nonnested systems of equations was applied, which involved reestimating theITSUR system and testing the relative performances of fitted values from each modelin a composite model. For nonnested models, there are typically four outcomes: rejectionof model 1, rejection of model 2, rejection of both models, and nonrejection of either(Greene). However, the Davidson and MacKinnon system "Pl" test has two rejection out-comes: rejection in the direction of the other model, and rejection away from the othermodel (see Davidson and MacKinnon, p. 304).

The P1 values in table 5 lead to a rejection of model 1 (a standard model) in the direc-tion of model 2 (technical inefficiency model). When reversing the test, model 2 wasrejected in the direction of model 1. However, the relative size of the test coefficients isrevealing. When model 1 is the null hypothesis, the relevant coefficient is 0.54, indicatinga sizeable move in the direction of model 2. When model 2 is the null hypothesis, therelevant coefficient is 0.11, indicating small movement in the direction of model 1.

Mixed results are not uncommon when applying nonnested tests. In this case, the non-nested systems P1 test leans in favor of model 2, which includes technical inefficiencyscores. A comparison of both models found more parameter estimates and t-statistics forimportant price variables were significant in model 2, the model that includes technicalinefficiency.

An effort also was made to test the specification of model 3. Because model 2 (technicalinefficiency model) is nested within model 3 (technical and allocative inefficiency model),a system log likelihood ratio was used to perform the test. The calculated X2 test statistic(6 degrees of freedom) of 11.0 indicated model 2 could not be rejected in favor of model3 at the 0.05 confidence level, but could be rejected at the 0.10 confidence level. Thus,if using the 0.05 confidence level, model 2 is the preferred model, but when using a 0.10confidence level, model 3 can be considered to be the preferable model.

Assuming model 3 is the preferred model, the supply parameters from this model canbe used to distinguish supply response among the different levels of technical and allo-cative inefficiency. Figure 1 shows the grain supply curves for two oblasts in 1994 whichproduced at a comparable level: Pskov and Nizhniy Novgorod. Using the parameter ofprice response for grain from model 3, output response was simulated assuming technicaland allocative efficiency, and again simulated at observed levels of inefficiency in these



u.3u

ILoNE 0.40a)U-

a 0.30

c 0.20

o 0.10-

80

0.00

100 120 140 160 180 200 220

1,000 tons

Standard ModelModel with Technical and Allocative Inefficiency

Figure 1. Hypothetical grain supply response under differentprice scenarios and efficiency levels: Nizhniy Novgorod andPskov oblasts, 1994

oblasts. When full technical and allocative efficiency are assumed, the model's estimatedoutput is greater over a range of prices around the observed mean relative price than

when inefficiency exists.In the case of Nizhniy Novgorod, the output bias would have been about 2.5% for prices

30% above the observed mean relative price; in the case of Pskov, the overstatement

would have been about 3.7% (figure 1). The percentage biases would have been larger

for the commodities with higher elasticities, such as sugar beets. It is important to note

that each oblast has a steeper supply slope in model 3 compared with model 1 because

of the interaction effect with price, which enters multiplicatively for technical inefficiency

[allocative inefficiency only changes the constant; see equation (16)]. The interpretation

is that a lower technical inefficiency score implies there will be a steeper, inelastic supply

curve because any price responsiveness is constrained by technical inefficiency. In figure

1, the slope of the supply curve from model 3 is steeper for Pskov, which was relatively

more technically inefficient. Aggregating these effects, the assumption of full efficiency

in each oblast in 1994 (compared to the sample average 0.76 score) means the predicted

grain output would have been overstated by about 2% if relative price levels had been

30% above observed price levels.

Conclusions

In this study we have demonstrated existing dual relationships can be used to directlyrelate technical and allocative inefficiency to the profit function, and the relationshipcan be used in estimating a system of output supplies and input demands. The method

presented here focuses on estimating the effect technical and allocative inefficiency

! /

Nizhniy Novgor Pskov

i I I I I

110 July 2002

n r- n ---------

I


scores have on output supply and input demand. Economists may consider using thisapproach in whole or in part. For example, if consistent data were available, economistscould take either existing technical inefficiency scores or existing allocative inefficiencyscores and estimate the influence of either of these scores on output supply and inputdemand. Or they could use this approach to model the joint influence of technical andallocative inefficiency on output supply and input demand.

This analysis focuses on incorporating inefficiency scores calculated with nonpara-metric methods into a system of output supply and input demand equations. Calculatinginefficiency scores using programming techniques has certain advantages. For example,programming methods impose no restrictions on functional forms and do not rely on non-linear econometric estimation techniques, which may have convergence problems. Pro-gramming methods can provide efficiency measures to use as regressors in econometricestimation. However, nonparametric methods do not readily lend themselves to estab-lishing confidence intervals around each inefficiency score.'Y The method introducedhere provides a tractable alternative to parametric approaches, which simultaneouslyestimate efficiency and the parameters of a profit function. This method allowsresearchers to gauge the effects of inefficiency on supply response and input use.

To illustrate this approach, efficiency scores were calculated in Russian agricultureand the scores were incorporated into an output supply and input demand system.Because Russia's agriculture was modeled over a period in which the economy was intransition, technical and allocative inefficiency were likely to exist. Our findings confirmtechnical and allocative inefficiencies were widespread. The Russian corporate farmswere shown to be slightly responsive to output prices, but more responsive to some ofthe input prices.

Perhaps in the future, efficiency scores calculated by programming approaches couldbe viewed as serving a role in cross-sectional econometric models analogous to the roleof time variables in time-series models. Or perhaps there may be useful analogs in otherfields of economics. For example, experimental economics has shown that consumersoften do not have perfect perceptions about price and quality of goods, so they do notengage in cost-minimizing or economically rational behavior. Another example is thecognitive dissonance experienced by researchers in the natural resources field whensoliciting consumer willingness-to-pay versus willingness-to-receive values for environ-mental amenities. These types of disparities might be translated into inefficiency scoresand used as variables in various demand models.

[Received August 2001; final revision received March 2002.]

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